PRODUCTION SYSTEMSENGINEERING

Chapter 11: Analysis of Exponential Lines

Instructors: J. Li (Univ. of Kentucky) and S. M. Meerkov (Univ. of Michigan)

Teaching Assistant: L. Zhang (Univ. of Michigan)

Copyright © 2008 J. Li, S.M. Meerkov and L. Zhang

Tsinghua University

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Motivation: The need for a quick and easy method for calculating PR,

TP, WIPi, BLi, STi in production lines with exponential machines.

Analysis of the effects of Tup and Tdown on the performance measures.

The approach is the same as in the Bernoulli case: direct analysis of two-machine lines and recursive aggregations for M > 2-machine systems.

The technicalities, however, are more involved.

1. Synchronous Exponential Lines

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1.1 Two machine case

1.1.1 Conventions

a) Blocked before service.

b) The first machine is never starved; the last machine is never blocked.

c) Flow model.

d) Machine states are determined independently from each other.

e) Time-dependent failures.

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1.1.2 State of the system

Triple (h, s1, s2):

h ϵ [0, N], si ϵ {up = 1, down = 0}

Boundary states:

(0, s1, s2) and (N, s1, s2)

Internal states:

(h, s1, s2), h ϵ (0, N)

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1.1.3 States pdf

Boundary states:

Internal states:

Clearly,

This pdf is calculated using methods of continuous time, mixed space Markov process.

1.1.4 Stationary probabilities of the boundary states

Here

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1.1.4 Stationary probabilities of the boundary states (cont.)

Lemma: Function Q(λ1, μ1, λ2, μ2) with λ1 ϵ (0, ∞) μ1 ϵ (0, ∞), and N ϵ (0, ∞), takes values on (0, 1) and isstrictly decreasing in λ2 and μ1,

strictly increasing in λ1 and μ2,strictly decreasing in N.

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1.1.5 Stationary marginal pdf of buffer occupancy

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Illustration: Lines with identical ei’s:

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1.1.5 Stationary marginal pdf of buffer occupancy (cont.)

1.1.5 Stationary marginal pdf of buffer occupancy (cont.)

Reversed lines with identical ei’s:

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1.1.5 Stationary marginal pdf of buffer occupancy (cont.)

Reversed lines with non-identical ei’s:

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1.1.6 Formulas for performance measures

Production rate

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1.1.6 Formulas for performance measures (cont.)

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Work-in-process

where

1.1.6 Formulas for performance measures (cont.)

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Blockages and starvations

i.e.,

(Recall that for Bernoulli lines, PR = p1 – BL1 = p2 – ST2.)

1.1.7 Effects of up- and downtime

Theorem: Consider lines l1 and l2, with machines of identical efficiency and finite buffers of identical capacity. Assume

Then,

This phenomenon is due to the fact that finite buffers accommodate shorter downtime easier than longer ones

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1.1.7 Effect of up- and downtime (cont.)

Note that for an isolated machine, increasing Tup by any factor or decreasing Tdown by the same factor has the same effect:

The situation is different for serial lines:

Theorem: In synchronous exponential two-machine lines defined by assumptions (a)-(e), PR has a larger increase when the downtime of a machine is decreased by a factor (1 + α), α > 0, than when the uptime is increased by the same factor.

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1.1.8 Asymptotic properties

Theorem:

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1.1.8 Asymptotic properties (cont.)

Illustration (for the six lines used in the illustration of pdf’s)

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1.2 M > 2-machine case

1.2.1 Mathematical description and aggregation preliminaries

Conventions: The same as in two-machine case. States:

Too complex – aggregation is used for simplification.

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1.2.1 Mathematical description and aggregation preliminaries (cont.)

Backward aggregation

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1.2.1 Mathematical description and aggregation preliminaries (cont.)

Forward aggregation

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1.2.2 Aggregation equations

with initial conditions:

and boundary conditions:

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1.2.2 Aggregation equations (cont.)

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Convergence

Theorem: All four sequences are convergent:

Moreover,

i.e.,

1.2.2 Aggregation equations (cont.)

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Interpretation of the limits:

i = 2,…, M – 1

1.2.3 Formulas for performance measures

Production rate:

Work-in-process

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1.2.3 Formulas for performance measures (cont.)

Blockages and starvations:

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1.2.4 PSE Toolbox

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1.2.5 Effects of up- and downtime

Remain the same as in two-machine exponential lines:

Shorter up- and downtime lead to larger than longer ones, even if the machine efficiencies remain the same.

Decreasing Tdown by any factor leads to a larger than increasing Tup by the same factor.

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1.2.6 Asymptotic properties

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1.2.6 Asymptotic properties (cont.)

Illustration (Line 1)

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1.2.7 Accuracy of estimates

is typically evaluated with the error within 1%. and have lower accuracy.

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1.2.8 System-theoretic properties

Theorem: Synchronous exponential lines are reversible:

Theorem: is strictly monotonically increasing in μi, i = 1,…, M, and Ni, i

= 1,…, M – 1; strictly monotonically decreasing in λi, i = 1,…, M.

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2. Asynchronous Exponential Lines

2.1 Two-machine case

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ci [parts/min], λi [1/min], μi [1/min]

TP [parts/min]

2.1.1 Mathematical description

Conventions (a)-(e) are assumed to hold.

States are the same as in synchronous lines.

Analysis is based on continuous time, mixed space Markov processes.

Calculations are a little more involved than in the synchronous case.

Function Q does not emerge in these calculations.

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2.1.2 Stationary marginal pdf of buffer occupancy

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where

2.1.2 Stationary marginal pdf of buffer occupancy (cont.)

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2.1.2 Stationary marginal pdf of buffer occupancy (cont.)

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2.1.2 Stationary marginal pdf of buffer occupancy (cont.)

Illustration

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2.1.2 Stationary marginal pdf of buffer occupancy (cont.)

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2.1.2 Stationary marginal pdf of buffer occupancy (cont.)

Observations remain the same as in the synchronous case:

Reversibility holds.

Larger up- and downtime qualitatively change pdf’s as compared with shorter ones.

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2.1.3 Formulas for performance measures

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Throughput:

2.1.3 Formulas for performance measures (cont.)

Work-in-process

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2.1.3 Formulas for performance measures (cont.)

Blockages and starvations

Represent

Denote

Then

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2.1.3 Formulas for performance measures (cont.)

Using conditional probability formula

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2.1.3 Formulas for performance measures (cont.)

Similar arguments are used to calculate BL1.

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2.1.4 Effects of up- and downtime

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Similar to those in the synchronous case:

Larger up- and downtime lead to lower TP than shorter ones.

It is more efficient to decrease Tdown than to increase Tup.

2.1.5 Asymptotic properties

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2.1.5 Asymptotic properties (cont.)

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2.1.5 Asymptotic properties (cont.)

Observations remain the same as in the synchronous case:

WIP may grow almost linearly in N, while TP is always saturating; thus, large N's are not advisable (see Chapter 14).

Reverse lines have identical TP.

Longer up- and downtime result in lower TP than shorter up- and downtime; in some cases, the difference is as large as 25%.

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2.2 M > 2-machine case

Approach: Aggregation procedure based on ci, bli, and sti.

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2.2.1 Aggregation equations

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with initial conditions

and boundary conditions

2.2.1 Aggregation equations (cont.)

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Convergence

Theorem: The aggregation procedure is convergent:

In addition,

2.2.1 Aggregation equations (cont.)

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Interpretation of and

i = 2,…, M – 1

2.2.2 Formulas for performance measures

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Throughput

or use the formulas for two-machine lines with and .

Work-in-process

For , use the two-machine formulas with and .

Blockages and starvations

2.2.3 PSE Toolbox

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2.2.4 Effects of up- and downtime

Remain the same as in synchronous lines.

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2.2.5 Asymptotic properties

→ min{ciei}

→ 0

→ cMeM –

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2.2.5 Asymptotic properties (cont.)

Illustration

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2.2.5 Asymptotic properties (cont.)

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2.2.6 Accuracy of the estimates

Lower than that for synchronous lines.

εTP is on the average within 5%.

εWIP is on the average within 8%.

εSTi and εBLi

is on the average within 0.03.

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3. Case Studies

3.1 Automotive ignition coil processing system

To account for the closed nature of the line, e1 and Tdown, 1 have been modified to 0.9226 and 19.12, respectively.

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3.1.1 Model validation

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3.1.2 Effect of starvation by pallets

Observation: No significant TP improvement if the closed line is unimpeding.

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3.1.3 Effect of increased buffer capacity and m9-10 efficiency

Observation: Substantial improvement (similar to that obtained using the Bernoulli description)

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3.2 Crankshaft production line (evaluation of the initial description)

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3.2.1 Layout

3.2.2 Structural model

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3.2.3 Machine and buffer parameters

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3.2.4 Performance and “what if” analysis

Sensitivity to buffer capacity

Sensitivity to Sta. 7 efficiency

Observation: Although the initial design meets specification, Sta. 7’s performance should be given a particular attention in order to maintain the production goal.

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4. Summary

The performance measures of serial lines with exponential machines can be evaluated using the same approach as in the Bernoulli case. Specifically, two-machines lines can be evaluated by closed-form expressions, and aggregation procedures can be used to analyze longer lines. However, all analytical expressions are more involved, especially in the asynchronous case.

Stationary marginal pdf's of buffer occupancy contain δ-functions, which account for buffers being empty and full.

The accuracy of the resulting performance measure estimates in the synchronous case is similar to that of the Bernoulli case. In the asynchronous case, the accuracy is lower.

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Shorter up- and downtime lead to a higher production rate (or throughput) than longer ones, even if machine efficiency remains constant.

A decrease of the downtime leads to higher throughput of a serial line than an equivalent increase of the uptime.

Exponential lines observe the usual monotonicity and reversibility properties.

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4. Summary (cont.)